Asymptotics in Spin-Boson Type Models

Abstract

In this paper, we investigate a family of models for a qubit interacting with a bosonic field. More precisely, we find asymptotic limits of the Hamiltonian as the interaction strength tends to infinity. The main result has two applications. First of all, we show that any self-energy renormalisation scheme similar to that of the Nelson model does not converge for the three-dimensional Spin-Boson model. Secondly, we show that excited states exist in the massive Spin-Boson model for sufficiently large interaction strengths. We are also able to compute the asymptotic limit of many physical quantities.

Various Transformation Statements

In this appendix various useful transformation theorems is stated. Sources are [4,5,6, 15] and [22].

Lemma A.1

Let U be a unitary operator from $\mathcal {H}$ into some Hilbert space $\mathcal {K}$. Then there is a unique unitary map $\varGamma (U):\mathcal {F}_b(\mathcal {H})\rightarrow \mathcal {F}_b(\mathcal {K})$ such that $\varGamma (U)\epsilon (g)=\epsilon (Ug)$. If $\omega$ is selfadjoint on $\mathcal {H}$, V is unitary and $f\in \mathcal {H}$ then

$$\begin{aligned} \varGamma (U)d\varGamma (\omega )\varGamma (U)^*&=d\varGamma (U\omega U^*).\\ \varGamma (U)W(f,V)\varGamma (U)^*&=W(Uf,UVU^*).\\ \varGamma (U)\varphi (f)\varGamma (U)^*&=\varphi (Uf). \end{aligned}$$

Furthermore, $\varGamma (U)(f_1\otimes _s\cdots \otimes _s f_n)=Uf_1\otimes _s\cdots \otimes _s Uf_n$ and $U\varOmega =\varOmega$.

Lemma A.2

Let $f,h\in \mathcal {H}$ and $U\in \mathcal {U}(\mathcal {H})$. Then

$$\begin{aligned} W(h,U)\varphi (g)W(h,U)^*&=\varphi (Ug)-2Re (\langle Ug,h \rangle )\\ W(h,U)a(g)W(h,U)^*&=a(Ug)-\langle Ug,h \rangle \\ W(h,U)a^\dagger (g)W(h,U)^*&=a^\dagger (Ug)-\langle h,Ug \rangle . \end{aligned}$$

Furthermore, if $\omega$ is a selfadjoint, non-negative and injective operator on $\mathcal {H}$ and $h\in \mathcal {D}( \omega U^* )$ then

$$\begin{aligned} W(h,U)d\varGamma (\omega )W(h,U)^*=d\varGamma (U\omega U^*)-\varphi (U\omega U^*h)+\langle h,U\omega U^*h\rangle \end{aligned}$$

on the domain $\mathcal {D}(d\varGamma (U\omega U^*))$.

In what follows we consider two fixed Hilbert spaces $\mathcal {H}_1$ and $\mathcal {H}_2$. We will need the following two lemmas.

Lemma A.3

There is a unique isomorphism $U:\mathcal {F}_b(\mathcal {H}_1\oplus \mathcal {H}_2)\rightarrow \mathcal {F}_b(\mathcal {H}_1)\otimes \mathcal {F}_b(\mathcal {H}_2)$ such that $U(\epsilon (f\oplus g))=\epsilon (f)\otimes \epsilon (g)$. If $\omega _i$ is selfadjoint on $\mathcal {H}_i$, $V_i$ is unitary on $\mathcal {H}_i$ and $f_i\in \mathcal {H}_i$ then

$$\begin{aligned} UW(f_1\oplus f_2,V_1\oplus V_2)U^*&=W(f_1,V_1)\otimes W(f_2,V_2)\\ Ud\varGamma (\omega _1\oplus \omega _2)U^*&=d\varGamma (\omega _1)\otimes 1 +1\otimes d\varGamma ( \omega _2)\\ U\varphi (f_1,f_2)U^*&=\varphi (f_1)\otimes 1+1\otimes \varphi (f_2)\\ Ua(f_1,f_2)U^*&=a(f_1)\otimes 1+1\otimes a(f_2)\\ Ua^{\dagger }(f_1,f_2)U^*&=a^{\dagger }(f_1)\otimes 1+1\otimes a^{\dagger }(f_2). \end{aligned}$$

Lemma A.4

There is a unique isomorphism

$$\begin{aligned} U:\mathcal {F}(\mathcal {H}_1)\otimes \mathcal {F}(\mathcal {H}_2)\rightarrow \mathcal {F}(\mathcal {H}_1)\oplus \bigoplus _{n=1}^{\infty } \mathcal {F}(\mathcal {H}_1)\otimes S_{n}(\mathcal {H}_2^{\otimes n}) \end{aligned}$$

such that

$$\begin{aligned} U(w \otimes \{\psi _2^{(n)} \}_{n=0}^\infty )=\psi ^{(0)}w\oplus \bigoplus _{n=1}^{\infty } w \otimes \psi _2^{(n)}. \end{aligned}$$

Let A be a selfadjoint operator on $\mathcal {F}(\mathcal {H}_1)$ and B be selfadjoint on $\mathcal {F}(\mathcal {H}_2)$ such that B is reduced by all of the subspaces $S_{n}(\mathcal {H}_2^{\otimes n})$. Write $B^{(n)}=B\mid _{S_{n}(\mathcal {H}_2^{\otimes n})}$. Then

$$\begin{aligned} U(A\otimes 1+1\otimes B)U^*&=A+B^{(0)}\oplus \bigoplus _{n=1}^\infty ( A\otimes 1+1\otimes B^{(n)} )\\ U A\otimes B U^*&=A \otimes B= B^{(0)} A \oplus \bigoplus _{n=1}^\infty A\otimes B^{(n)}. \end{aligned}$$